ZHANG Xi-hua, CHEN Yan-ni, ZHENG Yan-ping, DU Hong-ke
(College of Mathematics and Information Science, Shaanxi Normal University, Xi′an 710062, Shaanxi, China)
Abstract:
Eigenprojections at zero of Drazin invertible operators on a Hilbert space are studied. Two results are obtained. The first result is that if A is a Drazin invertible operator, then Q=Aπ if and only if Q2=Q, AQ=QA, σ(AQ)={0} and A+Q is invertible. The second one is that if E is an idempotent operator commuting with A and A is Drazin invertible with i(A)=k, then the following three statements are equivalent: E is the eigenprojection of A at 0; A+λE is invertible for all λ≠0; Ak E=0 and A+ξE is invertible for some ξ≠0.
KeyWords:
generalized inverse; Drazin inverse; eigenprojection; Drazin index
